MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Thermodynamics of Materials

3.00 Fall 2002

W. Craig Carter

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave.

Cambridge, MA 02139

Problem Set 8: Due Tues. Nov. 25, Before 5PM in in 13-5114

Exercise 8.1

Demonstrate that Poisson's ratio, $\nu$, cannot exceed $\frac{1}{2}$ by finding particular value of strain $\epsilon_{ij}$ (for an isotropic material with elastic Young's modulus $E_{el} > 0$) that makes the stored elastic strain energy negative.

Exercise 8.2

Starting with the Gibbs-Duhem expression for phases with fixed composition, derive the Clausius-Clapeyron relation $dP = (\Delta \ensuremath{\overline{S}}/\Delta \ensuremath{\overline{V}}) dT$.

Using a carefully worded sentence or two, describe what this Clausius-Clapeyron means physically.

Exercise 8.3

Consider a binary alloy with components $A$ and $B$, let $X^\alpha_A$, $X^\beta_A$, and $X^\gamma_A$ represent the compositions of three phases $\alpha$, $\beta$, and $\gamma$ that coexist at a triple point at $P = P_{tp}$ and $T = T_{tp}$.

Note that, for each phase in a binary alloy, the composition is given by one variable only because $X^\alpha_A = 1 - X^\alpha_B$, $X^\beta_A = 1 - X^\beta_B$, and $X^\gamma_A = 1 - X^\gamma_B$.

Starting with the Gibbs-Duhem expression, derive a relationship for the change in the triple point $dP_{tp} = ($material properties$) dT_{tp}$.

Also for the triple point, find a relationship between the change in the chemical potential of A ($d \mu_A$) and the change in the chemical potential of B ($d \mu_B$).

Exercise 8.4

In Homework problem 5.2, you found the equilibrium temperature and length of a thermally expanding in contact with a thermostat.

Using the engineering solution in the solution set, determine whether that stability is locally stable or unstable.